Two Model Abstraction Techniques Based on Temporal Grain Size: Aggregation and Mixed Models

Abstract: Many real world dynamic systems involve such a large number of variables and
interconnections that it is difficult to grasp them mentally in their
entirety. Abstracting a detailed description to produce a simpler description
becomes essential as the complexity of the subject system increases. For
example, an abstraction hierarchy of models is necessary to control the
combinatorial explosion of envisionment process [4]. The author has
investigated how different techniques for generating an abstract model from a
detailed dynamic model of a system. One technique is generation of a model of
a coarse temporal grain size from a model of a finer grain size by making
assumptions about the relative adjustment speeds of the equations in the model
[1]. The other technique is aggregating a dynamic system model to generate an
aggregate model when the original model is nearly decomposable [8]. This
paper compares the two abstraction techniques to show that they are actually
closely related.
The paper also discusses how the notion of causality relates to that of model
abstraction. Both abstraction techniques are means of going bottom-up from a
detailed description of a system to a description at a higher level of
abstraction. There is an alternative, top-down, way of looking at the
situation. When we model a complex system, we carry our modeling only down to
some level of components. Above that level, structure and the interrelations
of components are explicit. Below that level, the components are black boxes
with no detailed internal structure. Suppose that we determine the causal
structure of the model but decide subsequently that some part of the model
must be elaborated in greater detail. Will the new model, incorporating this
elaboration, have the same causal structure as the more aggregated model, or
do we have to reexamine the causal ordering from the beginning? Section 2
will show that it is not necessary to reexamine the causal ordering of the
aggregated model after elaborating a part of an aggregated model.
Kuipers uses abstraction by time-scale in order to control the exponential
growth of the number of possible courses of behavior in qualitative simulation
[4]. Kuipers has a hierarchy of constraint networks of very fast to very slow
mechanisms. When simulating a fast mechanism, variables controlled by slower
mechanisms are considered constant, and when simulating a slow mechanism,
equilibrium among variables controlled by faster mechanisms is considered to
be reached instataneously. This idea of abstraction by time-scale is similar
to the notion of abstraction discussed in this paper. The abstraction
techniques discussed in this paper can be used to generate a hieracrchy of
models of different time scales.